From Chang and Keisler's "Model Theory", section 7.2, we know that:
There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^+,\alpha)$ iff there exists a tree $T$ of height $\alpha^+$, with at most $\alpha$ elements at each level $\xi<\alpha^+$, and with no branch of length $\alpha^+$.
There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^{++},\alpha)$ iff there is a (Kurepa) family $F$ of subsets of $\alpha^+$ such that $|F|=\alpha^{++}$ and for every $\xi<\alpha^+$, $\lvert\{X\cap\xi\mid X\in F\}\rvert=\alpha.$
My question: Are there any "natural" statements that would be equivalent to a sentence $\sigma$ admitting $(\alpha^{+n},\alpha)$, $3\le n<\omega$? For Chang and Keisler, $\sigma$ has to be first-order. For our purposes, even $L_{\omega_1,\omega}$ is good enough.
Addition: Definition A sentence $\sigma$ in a language with a unary predicate $P$ admits $(\kappa,\lambda)$, if $\sigma$ has a model $M$ such that $\lvert M\rvert = \kappa$ and $\lvert P^M\rvert=\lambda$. Of course, $\lambda\le\kappa$. So, not only we need a model of a specific size, but we need the predicate $P$ to have a specific size too.
